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Let $K<X> =K<X_1,...,X_n>$ be the free $K$-algebra on $X={X_1,...,X_n}$ overa field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each$X_i$ is assigned a positive degree), and let ${\cal G}$ be a finitehomogeneous Gr\"obner basis for the ideal $I=<{\cal G}>$ of $K<X>$ with respectto some monomial ordering $\prec$ on $K<X>$. It is proved that if the monomialalgebra $K<X>/<{\bf LM}({\cal G})>$ is semi-prime, where ${\bf LM}({\cal G})$is the set of leading monomials of ${\cal G}$ with respect to $\prec$, then the$\mathbb{N}$-graded algebra $A=K<X>/I$ is semiprimitive (in the sense ofJacobson). In the case that ${\cal G}$ is a finite non-homogeneous Gr\"obnerbasis with respect to a graded monomial ordering $\prec_{gr}$, and the$\mathbb{N}$-filtration $FA$ of the algebra $A=K<X>/I$ induced by the$\mathbb{N}$-grading filtration $FK<X>$ of $K<X>$ is considered, if themonomial algebra $K<X>/<{\bf LM}({\cal G})>$ is semi-prime, then it is provedthat the associated $\mathbb{N}$-graded algebra $G(A)$ and the Rees algebra$\widetilde{A}$ of $A$ determined by $FA$ are all semiprimitive.

Recognizing The Semiprimitivity of $\mathbb{N}$-graded Algebras via Gr\"obner Bases